The rest of the paper is organised as follows. Section 2 briefly recaps the formulation of viscoelastic channel flow described in our earlier work [1]}. Section 3 then outlines the branch continuation approach, with the technical details relegated to a series of Appendices. The results are presented in two sections: section 4 considers finite inertia \(Re >0\)  and section 5 deals with inertialess flows at \(Re=0\) . Section 4 exclusively concentrates on \(\beta =0.9\)  and considers how the subcritical travelling wave branches behave as: 1) \(Re\)  varies over the range \(Re \in [0,3000]\) ; and 2) as the domain size varies at \(Re=30\) .
For the analysis of creeping flow in section 5 at \(Re=0\) , we explore the existence of the travelling waves over the \((Wi,\beta )\)  plane for \(\beta \in [0.5,1)\)  and \(Wi < 50\)  and \(L_{max} \in \lbrace 70,100,500\rbrace \) , the maximum polymer extensibility in the FENE-P model.
[2]} has concurrently found travelling waves in viscoelastic channel flow at \(Re=0.01\)  by the complementary approach of time stepping in the Phan-Thien-Tanner model.
These waves correspond to the attracting upper branch of the curves shown here. Finally a discussion follows in section 6.
